Hyper-luminous quasars (L_bol ≳ 10⁴⁷ erg s⁻¹) are ideal laboratories to study the interaction and impact of the extreme radiative field and the most powerful winds in the active galactic nuclei (AGN) nuclear regions. They typically exhibit low coronal X-ray luminosity (L_X) compared to the ultraviolet (UV) and mid-infrared (MIR) radiative outputs (L_UV and L_UV); a non-negligible fraction of them report even ∼1 dex weaker L_X compared to the prediction of the well established L_X–L_UV and L_X–L_UV relations followed by the bulk of the AGN population. In our WISE/SDSS-selected Hyper-luminous (WISSH) z = 2 − 4 broad-line quasar sample, we report on the discovery of a dependence between the intrinsic 2–10 keV luminosity (L_{2 − 10}) and the blueshifted velocity of the CIV emission line (v_CIV) that is indicative of accretion disc winds. In particular, sources with the fastest winds (v_CIV ≳ 3000 km s⁻¹) possess ∼0.5–1 dex lower L_{2 − 10} than sources with negligible v_CIV. No similar dependence is found on L_UV, L_UV, L_bol, the photon index, or the absorption column density. We interpret these findings in the context of accretion disc wind models. Both magnetohydrodynamic and line-driven models can qualitatively explain the reported relations as a consequence of X-ray shielding from the inner wind regions. In case of line-driven winds, the launch of fast winds is favoured by a reduced X-ray emission, and we speculate that these winds may play a role in directly limiting the coronal hard X-ray production.